5.4 Analyzing Graphs of Polynomial Functions Honors Algebra 2 Mrs. Pam Miller

Graphing a Polynomial Function We will NOT graph by hand….just on our calculators! f(x) = -x4 + x3 + 3x2 + 2x Is this an even or odd degree function? Is the leading coefficient positive or negative? Describe the end behavior of the function. NOW, graph the function on your calculator . How many REAL zeros does this function have? How many imaginary solutions?

Analyzing sign changes to “locate” REAL zeros. Location Principle Analyzing sign changes to “locate” REAL zeros. Look at the TABLE of ordered pairs that make up the graph of this polynomial function. Look for sign changes in the “y” values. Where do the REAL zeros occur?

You Try… Determine the consecutive integer values of x between which each real zero of this function is located. f(x)= x4 – x3 – 4x2 + 1

Maximum & Minimum Points This graph is the general shape of a 3rd degree polynomial function. The maximum & minimum values of a function are called the EXTREMA. These points are often referred to as TURNING POINTS. The graph of a polynomial function of degree n has at most n – 1 turning points.

Find the Maximum & Minimum Use your calculator to find the relative maximum. The relative maximum is 5 & it occurs at x = 0.

Find the Maximum & Minimum Use your calculator to find the relative minimum . The relative minimum is 1 & it occurs at x = 2.

The graph of a polynomial function will have interval(s) over which it is increasing and interval(s) over which it is decreasing. Increasing Intervals: Decreasing Interval:

Graph: 2x3 + x2 - 4x - 2 Answer these questions based on the graph. Find the relative maximum & the x-coordinate at which it occurs. Find the relative minimum & the x-coordinate at State the Domain . State the Range. State the consecutive integer values of x between which each real zero is located. Find the actual real zero(s) rounding to nearest hundredth as necessary. 7) State the interval(s) over which the function is increasing/decreasing. (-1, 1) (.67, -3.63) All Real Numbers All Real Numbers Between : -2 & -1; -1 & 0; 1 & 2 -1.41, -.5, 1.41 Increasing: x < -1; x > .67 Decreasing: -1< x < .67

Graph: f(x) = x4 – x3 – 4x2 + 1 Answer these questions based on the graph. Find the relative maximum & the x-coordinate at which it occurs. Find the relative minima & the x-coordinate at which each occurs. State the Domain . State the Range. 5) How many REAL zeros exist? How many imaginary? 6) State the consecutive integer values of x between which each real zero is located. Find the actual real zero(s) rounding to nearest hundredth as necessary. Describe the end behaviors of the graph . 9) State the interval(s) over which the function is increasing/decreasing. 1 at x = 0 -1.05 at x = -1.09 & -7.31 at x = 1.84 All REAL Numbers {y|y ≥ -7.31} 4 Between: -2 & -1; -1 & 0; 0 & 1; 2 & 3 - 1.44, -.57, .49, 2.52 x → + ∞, f(x) → + ∞; x → - ∞, f(x) → + ∞ Decreasing: x < -1.09; 0 < x < 1.84; Increasing: -1.09 < x < 0; x > 1.84